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Lower bounds on the complexity of polytope range searching. (English) Zbl 0695.68032

The range search problem of size (n,p) is specified by a set P of n points in Euclidean d-space and a collection Q of subsets \(q\subseteq {\mathbb{R}}^ d\), called queries, with \(| \{P\cap q:\) \(q\in Q\}| =p\). Here Q may be the set of all hyperrectangles, simplices, balls in Euclidean d-space, etc. Suppose we are given m units of computer memory. The problem is to organize the memory to be a position to answer the following questions efficiently: Given an arbitrary query \(q\in Q\), how many of the n points lie inside q? The author proves a family of lower bounds on the space-time complexity of the range search problem where Q is a collection of simplices or slabs.
First, it is proved that any range search problem of size (n,p) may be solved within \(t=O(n/\log_ 2(p/n))\) query time and \(m=O(p/\log_ 2(p/n))\) memory cells. Surprisingly, this result is in fact optimal: there exists a constant \(c>0\) and a class of search problems of size (n,p) that require \(m=\Omega (p)\) memory cells to be solved in query time \(t\leq cn/\log_ 2(p/n).\)
Second, it is proved that simplex range searching on n points requires \(\omega\) (n/\(\sqrt{m})\) query time in two dimensions and \(\Omega ((n/\log_ 2n)/m^{1/d})\) query time in any dimension \(d\geq 3.\)
These bounds hold for a random point-set with high probability, and thus are valid in the worst case as well as on the average. Interestingly, they still hold if the queries are restricted to congruent copies of a fixed simplex or even a fixed slab. The paper also contains several results of independent interest regarding to bipartite Ramsey theory and an intriguing generalization of Heilbornn’s problem: Given two integers n and \(k\leq n\), place n points in a unit square so that the convex hull of any k of them has an area at least \(\Omega\) (k/n). It is shown that this can be done if k exceeds \(\log_ 2n\).
Reviewer: S.P.Yukna

MSC:

68Q25 Analysis of algorithms and problem complexity
52Bxx Polytopes and polyhedra
68U99 Computing methodologies and applications
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